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arXiv:1008.4243 · Physical Review A

Quantifying non-Gaussianity for quantum information

Marco G. Genoni, Matteo G. A. Paris

Published Nov 30, 2010• 201 citations
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Abstract

We address the quantification of non-Gaussianity (nG) of states and operations in continuous-variable systems and its use in quantum information. We start by illustrating in detail the properties and the relationships of two recently proposed measures of nG based on the Hilbert-Schmidt distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behavior in most of the examples taken into account. However, we also show that they introduce a different relation of order; that is, they are not strictly monotone to each other. We exploit the nG measures for states in order to introduce a measure of nG for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in detail the role played by nG in entanglement-distillation protocols. Besides, we exploit the QRE-based nG measure to provide different insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information, and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE nG to the quantum Fisher information. Finally, since evaluation of the QRE nG measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to nG for some classes of states and by considering the possibility of performing on the states only certain efficient or inefficient measurements.

DOI
10.1103/physreva.82.052341

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We address the quantification of non-Gaussianity (nG) of states and operations in continuous-variable systems and its use in quantum information. We start by illustrating in detail the properties and the relationships of two recently proposed measures of nG based on the Hilbert-Schmidt distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behavior in most of the examples taken into account. However, we also show that they introduce a different relation of order; that is, they are not strictly monotone to each other. We exploit the nG measures for states in order to introduce a measure of nG for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in detail the role played by nG in entanglement-distillation protocols. Besides, we exploit the QRE-based nG measure to provide different insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information, and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE nG to the quantum Fisher information. Finally, since evaluation of the QRE nG measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to nG for some classes of states and by considering the possibility of performing on the states only certain efficient or inefficient measurements.
We introduce a measure to quantify the non-Gaussian character of a quantum state: the quantum relative entropy between the state under examination and a reference Gaussian state. We analyze in detail the properties of our measure and illustrate its relationships with relevant quantities in quantum information such as the Holevo bound and the conditional entropy; in particular, a necessary condition for the Gaussian character of a quantum channel is also derived. The evolution of non-Gaussianity is analyzed for quantum states undergoing conditional Gaussification toward twin beams and de-Gaussification driven by Kerr interaction. Our analysis allows us to assess non-Gaussianity as a resource for quantum information and, in turn, to evaluate the performance of Gaussification and de-Gaussification protocols.
We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a reference Gaussian state. We analyze in detail the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single-mode and multimode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows one to define a corresponding measure for the non-Gaussian character of a quantum operation.
Nonclassical states of quantized light, except for the Gaussian states, also possess non-Gaussian phase-space distributions. Despite several attempts, a unified description of nonclassicality (NC) and non-Gaussianity (NG) of quantum states of light has not been developed as-of-yet. Here, we propose an experimentally verifiable scheme for quantification of NC, in terms of Wehrl entropy, that further leads to the simultaneous quantification of NC and NG. While requiring no optimization, present work recovers earlier results qualitatively as well as explores several new possibilities on the conjugation of nonclassical and non-Gaussian character of quantum states. Moreover, current formalism, due to its possible extension to the finite-dimensional systems, bridges the gap between discrete and continuous variable systems. Our work, thus, becomes crucial in describing NC of quantum processes including open quantum systems as well as understanding the role of NC and NG as resources in several information theoretic tasks processing such as entanglement distillation, quantum network, quantum computation etc.
Non-Gaussianity of quantum states is a very important source for quantum information technology and can be quantified by using the known squared Hilbert–Schmidt distance recently introduced by Genoni et al. (Phys. Rev. A 78 042327 (2007)). It is, however, shown that such a measure has many imperfects such as the lack of the swapping symmetry and the ineffectiveness evaluation of even Schrödinger-cat-like states with small amplitudes. To deal with these difficulties, we propose an improved measure of non-Gaussianity for quantum states and discuss its properties in detail. We then exploit this improved measure to evaluate the non-Gaussianities of some relevant single-mode non-Gaussian states and multi-mode non-Gaussian entangled states. These results show that our measure is reliable. We also introduce a modified measure for Gaussianity following Mandilara and Cerf (Phys. Rev. A 86 030102(R) (2012)) and establish a conservation relation of non-Gaussianity and Gaussianity of a quantum state.
We consider how to quantify non-Gaussianity for the correlation of a bipartite quantum state by using various measures such as relative entropy and geometric distances. We first show that an intuitive approach, i.e., subtracting the correlation of a reference Gaussian state from that of a target non-Gaussian state, fails to yield a non-negative measure with monotonicity under local Gaussian channels. Our finding clearly manifests that quantum-state correlations generally have no Gaussian extremality. We therefore propose a different approach by introducing relevantly averaged states to address correlation. This enables us to define a non-Gaussianity measure based on, e.g., the trace-distance and the fidelity, fulfilling all requirements as a measure of non-Gaussian correlation. For the case of the fidelity-based measure, we also present readily computable lower bounds of non-Gaussian correlation.
We address the non-Gaussianity (nG) of states obtained by weakly perturbing a Gaussian state and investigate the relationships with quantum estimation. For classical perturbations, i.e. perturbations to eigenvalues, we found that the nG of the perturbed state may be written as the quantum Fisher information (QFI) distance minus a term depending on the infinitesimal energy change, i.e. it provides a lower bound to statistical distinguishability. Upon moving on isoenergetic surfaces in a neighbourhood of a Gaussian state, nG thus coincides with a proper distance in the Hilbert space and exactly quantifies the statistical distinguishability of the perturbations. On the other hand, for perturbations leaving the covariance matrix unperturbed, we show that nG provides an upper bound to the QFI. Our results show that the geometry of non-Gaussian states in the neighbourhood of a Gaussian state is definitely not trivial and cannot be subsumed by a differential structure. Nevertheless, the analysis of perturbations to a Gaussian state reveals that nG may be a resource for quantum estimation. The nG of specific families of perturbed Gaussian states is analysed in some detail with the aim of finding the maximally non-Gaussian state obtainable from a given Gaussian one.